The null hypothesis in analysis of
variance is that all samples (e.g., the samples of students with secure,
fearful, preoccupied, and dismissing attachment from Dr. Bliwise's data set)
came from populations that have the same mean (in this case, the same mean
score on the interpersonal anxiety measure). The test is called analysis of
variance and not analysis of means because it actually uses variance
calculations. You can think about it as a test that examines the variation
among the means: Are all the means very similar to each other or do they
differ significantly.

In class, Dr. Bliwise used the LifesaversÒ
example to demonstrate the null hypothesis. Lifesavers are a kind of candy
that comes in several different colors, each of which has its own flavor
(e.g., orange, cherry, pine-apple, and so on.) In this example, there were
four colors/flavors. The company that makes Lifesavers uses the same
solution for all of them and then adds flavorless scents that combine with
the generic flavor to give consumers the illusion that they experience
different tastes. Based on this knowledge, you would predict that people
would have a difficulty trying to guess the color of a Lifesaver with their
eyes and noses closed tight, since the only remaining information would be
the taste, which is identical in the absence of scents. Dr. Bliwise asked
each student in the class to try and guess the color of a Life Saver with
eyes and nose closed tight, and she predicted that they would essentially be
guessing at random without the ability to smell the identifying essence.
Thus, assuming that there were equal numbers of the four different kinds of
Lifesavers, 25% of the students would guess right simply by chance. We can
divide the students into 4 groups, depending on the color they named (e.g.,
red, green, blue, and yellow). Under the null hypothesis, each
group would have gotten equal numbers of the four kinds of Lifesavers. In
other words, the four groups did not differ in the composition of
Lifesavers.

If we apply the above example to the question
about the four attachment styles and mean interpersonal anxiety, then we
could think of the four kinds of Lifesavers as different levels of anxiety.
Under the null hypothesis, the four samples of students (four different
attachment styles) do not differ in their composition of anxiety levels
(their mean anxiety is the same).

The alternate hypothesis is that there
is a difference somewhere among the means of the samples. For example, if
all students cheated on the Lifesavers exercise by not closing their eyes or
noses, or if the Lifesavers did indeed have different tastes, then the
composition of Lifesavers across the four groups of students would differ.
Students who names their Lifesaver red, green, blue, or yellow would have
actually had a red, green, blue, or yellow piece of candy. Notice, however,
that all four groups do not have to differ for the null hypothesis to be
false. For example, in the case of attachment style and interpersonal
anxiety, we might predict that students with secure and dismissing
attachment have similar (low) levels of anxiety and that they differ
significantly from students with fearful and preoccupied attachment (high
anxiety). The omnibus ANOVA F-test tells you that there is a difference
somewhere (that the null hypothesis is false) but it does not tell you the
nature of the difference.

Notice that in the first example, where the
null hypothesis is true, the variability of Lifesavers (or anxiety levels)
within each group was the same as the variability of Lifesavers
between the groups. The extent to which there were different kinds of
Lifesavers within each group roughly corresponds to the within-group
estimate of the population variance. The extent to which the composition
of Lifesavers in each group differed from the composition of the other
groups corresponds to the between-group variance. (Technically, it is
variance among several groups and should be called among-groups
variance, but traditionally the word "between" has been used.)
When the null hypothesis is true, the variance within groups
equals the variance between groups. This is so, because the means of
the groups differ by chance (due to sampling error) much like the individual
scores within the groups differ by chance (due to the same sources of
sampling error). Notice that in the second example, where the null
hypothesis was false, the groups differed from each other more so than did
the Lifesavers within each group. Since students who said "red" tended to
have mostly red Lifesavers, the variability of Lifesavers within the "red"
group was very small. Since students who said "blue" tended to have mostly
blue Lifesavers and did not have blue, green, or yellow ones, the
variability within the "blue" group was small. The variability among the
groups (between-group variance) is much larger, because "mostly red" is very
different from "mostly blue." If you think of the different Lifesaver colors
as different levels of interpersonal anxiety, then anxiety scores were
similar within each attachment group, and then were different across the
groups. Thus, a group with mostly low anxiety scores would have a low mean
score, whereas a group with mostly high anxiety scores would have a high
mean score, and the difference (variance) among or between group
means would be larger than the variance among individual scores within
the groups.

Thus, F = (between-group variance
estimate)/(within-group variance estimate)When F<1 or F=1, between-group variance is smaller than within group
variance, suggesting that the groups differ as much as one would expect due
to sampling error. When F >> 1, then we gain confidence to say that there is
something going on, because the variability due to sampling error
(within-group variance) is much smaller than the variability among the
groups. Something else is going on: peeking or sniffing in the Lifesavers
example, or real differences in anxiety across attachment styles. The
between-group and within-group estimates of variance are calculated based on
different information from the sample. This tutorial does not cover the
calculations. The F-statistic has 2 kinds of degrees of freedom
- one for the between-group estimate and one for the within-group estimate.Dfbetween = k-1, where k=number of groups.Dfwithin = n-k, where n = total sample size.The sampling distribution of F is called and F-distribution.
Once you look-up the F critical value using the alpha-level and the
degrees of freedom, you can compare it to the F-observed. Variances
are always positive, and therefore F is always > 0. Hypotheses for F are
always non-directional.